Multivariable Calculus

Transcription

1 Multivariable Calculus Geometric Representation of Functions When we study functions from R to R, we find it useful to visualize functions by drawing their graphs When we are concerned with functions of two variables, ie from R 2 to R this technique is even more useful As we needed two dimensions to draw the graph of a function from R to R, we need three dimensions to draw the graph of a function from R 2 to R When drawing graphs of unfamiliar functions from R to R in two dimensions, we could mark out a number of points to get an idea of what the function looks like When we have a function from R 2 to R, we can do the same, but because of the extra dimension, we need a lot more points to build up a graph of the function One more systematic way to proceed is to draw the usual one dimensional graphs on various two-dimensional slices or cross-sections and then put these together to draw the graph Example 1 Say we are trying to graph the function f : R 2 R given by z f(x, y) x 2 y 2 for all (x, y) R 2 We can take a slice at in the y b planes In this plane z x 2 b 2 which is a parabola shifted down by b 2 units We draw these cross-sections for several values of b Next consider a cross-section in the plane x 0 The restriction to this plane is the upside-down parabola z y 2 Finally attach each of your cross-sections in the y b plane to your cross-section in the x 0 plane and sketch the rest Cross-Section Applet Another way to visualize a function from R 2 to R is to use level curves This technique only requires two-dimensional sketching For each (x, y) we evaluate f(x, y) to obtain, say b Then we draw the locus of points (x, y) in the xy-plane for which f has the same value b

2 Definition Let f : R n R be a function and let b R A level set is a set of the form {(x 1,, x n ) R n f(x 1,, x n ) b} When n 2,we call a level set a level curve, when n 3 we call it a level surface and when n 3 In general we may also call a level set a level hypersurface Example 2 Consider the function given by f(x, y) x 2 y 2 Start with the point (0, 0) at which f takes value 0 Now find all points where f is 0 This is the set {(x, y) x 2 y 2 0}, which is simply the inverse image of the set {0} ie f (0) This level curve is the set of all points for which x 2 y 2 ie such that x ±y Now consider the point (2, 1) at which f takes a value of 3 The level curve in this case is f (3) {(x, y) x 2 y 2 3}, which is a hyperbola centred on the origin with asymptotes y x and y x and focal points 3 and 3 Compute more level sets for different b, say 9, 5, 3, 5 and 9 Draw them in the xy-plane and then think of pulling each level curve up into the z b plane y f ( 9) f ( 3) f (0) f ( 5) f (3) f (5) f (9) x Figure 1: Level curves of the function given by f(x) x 2 y 2 Level Curve Applet 2

3 Example 3 Level sets are common in economics An indifference curve is just a level curve of the utility function Consider a utility function u : R 2 R given by u(x, y) Then for any two bundles (x 1, y 1 ) and (x 2, y 2 ) on the same level curve the consumer is indifferent, because u(x 1, y 1 ) u(x 2, y 2 ) An isoquant is a level curve of a production function Take a simple Cobb- Douglas production function given by Q KL, where K and L are quantities of inputs, say captial and labour and Q is the amount output produced from the inputs To draw an isoquant for Q 10, solve the equation KL 10 for L in terms of K and then graph the result Solving we find that the isoquant for Q 10 is a branch of a hyperbola L 10/K Do this for several values of Q, to build up a picture of the production function So far we have looked at drawing graphs for functions from R or R 2 to R We can also draw picture of functions from R to R 2 or R 3 A typical function from R to R 2 would be written as x(t) (x 1 (t), x 2 (t)), where are x 1 and x 2 are the coordinate functions of x For each t, x(t) is a point in R 2 By marking each such point x(t) in the x 1, x 2 plane, we trace out a curve in the plane This curve is the image of x v p + tv p Figure 2: The parameterized line x(t) (p 1 + tv 1, p 2 + tv 2) x 2 x 1 x 3 Figure 3: The parameterized curve x(t) (cos t, sin t, t) 3

4 Special Kinds of Functions Definition A linear function from R k to R m is a function f that preserves the vector space structure, ie f(x + y) f(x) + f(y) and f(λx) λf(x) for all x, y R k and all λ R Linear functions are sometimes called linear transformations Example 4 An example of a linear function is the function f : R k R given by for some a R k f(x) a x a 1 x a k x k, It turns out that every linear real-valued function is of the form given in the example above Theorem 1 Let f : R k R be a linear function Then, there exists a vector a R k such that f(x) a x for all x R k Thus every real-valued function on R k can be written as f(x) a x ( ) a 1 a k x 1 x k The level sets of a linear function into R are the sets a x b which we called hyperplanes when we were looking at vectors Theorem 2 Let f : R k R m be a linear function Then, there exists an m k matrix A such that f(x) Ax for all x R k This says there that every linear function from R k to R m can be associated with an m k matrix A Definition A quadratic form on R n is a real-valued function of the form n Q(x) a ij x i x j x T Ax i,j1 where A is any symmetric n n matrix Example 5 The general two-dimensional quadratic form a 11 x a 12 x 1 x 2 + a 22 x 2 2 can be written as Note the symmetry of A x T Ax ( x 1 x 2 ) ( a 11 a 12 a 12 a 22 ) ( x1 x 2 ) 4

5 Linear functions and quadratic forms are special forms of class of functions called polynomials which are functions made up of the sum of monomials Definition A function f : R k R is called a monomial if it is of the form f(x) cx a1 1 xa2 2 xa k k where c R and a 1,, a k are nonnegative integers The sum of the exponents a a k is called the degree of the monomial Example 6 1 f(x 1, x 2 ) 6x 3 1x 2 is a monomial of degree four 2 g(x 1, x 2, x 3 ) 2x 1 x 2 x 3 3 is a monomial of degree five 3 A constant function is a monomial of degree zero Definition A function f : R k R is called a polynomial if it is a finite sum of monomials on R k The highest degree of these monomials is called the degree of the polynomial A function f : R k R m is called a polynomial if each of its coordinate functions is a real-valued polynomial Example 7 1 f(x 1, x 2, x 3 ) 2x 2 1x 3 3 5x 1 x 2 x x 8 2x 3 is a polynomial of degree nine 2 A linear real-valued function is a polynomial of degree one 3 A quadratic form is a polynomial of degree two Definition A function f : R k R m is called an affine function if it is of the form f(x) Ax + b, where A is an m k matrix and b R m So an affine function is a polyonomial of degree one, and each component of f has the form Example 8 f i (x) a i1 x 1 + a i2 x a ik x k + b i a i x + b i 1 f(x) 2x + 1 is an affine function ( ) ( ) ( 1 4 x1 2 2 g(x 1, x 2 ) x 2 6 affine function ) ( ) x1 + 4x is an 2x 1 + 3x 2 5

6 Continuous Functions Just as we defined continuity for functions from subsets of R to R, we can define continuity for functions from subsets of R k to R m Again, the idea is that as the vector x gets near but not equal to x 0, the value of the function at x gets close to f(x 0 ) Definition Let f be a function into R m whose domain is a subset of R k The function f is continuous at x 0 in dom(f) if, for every sequence (x n ) in dom(f) converging to x 0, we have lim f(x n ) f(x 0 ) If f is continuous at each point of a set S dom(f), then f is said to be continuous on S The function f is said to be continuous if it is continuous on dom(f) Again, we can formulate an ε-δ definition of continuity Because of the higher dimension, we use ε-balls instead of ε-intervals in the definition Theorem 3 (ε-δ definition of continuity) Let f be a function into R m whose domain is a subset of R k Then f is continuous at x 0 dom(f) iff for each ε > 0 there exists δ > 0 such that x dom(f) and x B δ (x 0 ) imply f(x) B ε (f(x 0 )) For a function into R in two variables x and y this says: if we draw two xyplanes, no matter how close together, we can always cut off a cylinder such that all that part of the surface which is contained in the cylinder lies between the planes Continuity Applet Definition Let f, g : R k R m be functions and let c R We define new functions from R k into R m as follows cf given by (cf)(x) cf(x) (cf 1 (x),, cf m (x)); f + g given by (f + g)(x) (f 1 (x) + g 1 (x),, f m (x) + g m (x)); fg given by (fg)(x) (f 1 (x)g 1 (x),, f m (x)g m (x)); for all x R k Theorem 4 Let f, g : R k R m be functions that are continuous at x 0 R k and let c R Then 1 cf is continuous at x 0 ; 6

7 2 f + g is continuous at x 0 ; 3 fg is continuous at x 0 The following theorem follows from the sequential definition of continuity and the fact that a sequence in R m converges iff each of the m component sequences converges in R It can be used, together with our theorem about continuity of combinations of real-valued functions from R, to prove the previous theorem Theorem 5 Let f R k R m be a function Then, f is continuous at x 0 R k iff each of its coordinate functions f i : R k R is continuous at x 0 Theorem 6 If f is continuous at x 0 R k and g is continuous at f(x 0 ) R m, then the composite function g f is continuous at x 0 A partial derivative of a function of several variables is its derivative with respect to one of those variables with the others held constant Definition Let f : R n R be a function and let a (a 1,, a n ) R n We say that f has a partial derivative with respect to x i at a if the limit f(a 1,, a i + h,, a n ) f(a 1,, a n ) lim h 0 h exists and is finite We write (/ x i ) for the partial derivative of f at a with respect to x i : f(a 1,, a i + h,, a n ) f(a 1,, a n ) lim x i h 0 h Sometimes we write f i, f xi or D i f to denote the partial derivative with respect to x i at a Example 9 Consider the function f given by f(x, y) 3x 3 + 2x 2 y 2xy 2 Partial Derivatives Applet The Total Derivative Suppose we are interested in the behaviour of a function f(x, y) of two variables in the neighbourhood of some point (x, y ) If we hold y fixed at y and change x to x + x, then f(x + x, y ) f(x, y ) x (x, y ) x Similarly, if we hold x fixed at x and change y to y + y, then f(x, y + y) f(x, y ) y (x, y ) y 7

8 Since we are working with linear approximations, we can add the effects of the one-variable changes to find the approximate effect of a simultaneous change in x and y: f(x + x, y + y) f(x, y ) x (x, y ) x + y (x, y ) y Often we write f(x + x, y + y) f(x, y ) + x (x, y ) x + y (x, y ) y To interpret this, consider the approximation for a one-variable function f(x + h) f(x ) + f (x )h f(x ) + f (x )h f(x + h) x x + h Figure 4: The tangent line to the graph of f at x is a good approximation of the graph in the vicinity of (x, f(x ) For a function f(x, y) of two variables, the graph is a two-dimensional surface in R 3 and the analogue of the tangent line is the tangent plane to the graph We will show that f(x + x, y + y) f(x, y ) + x (x, y ) x + y (x, y ) y, states that the tangent plane to the graph at the point p (x, y, f(x, y )) is a good approximation to the graph near p Recall that to compute parameterized equation of the tangent plane P through the point p, we need two independent vectors u and v in the plane In this case we parameterize the plane as {x R 3 x p + su + tv for some s, t R} u (1, 0, / x(x, y )) and v (0, 1, / x(x, y )) are two independent vectors in the tangent plane 8

9 x u z p v y z z p p v 1 x x 1 u y y y (x, y ) y x (x, y ) x Figure 5: The tangent plane to the graph of f at p (x, y ) is a good approximation of the graph in the vicinity of (x, y, f(x )) Tangent Plane Applet Thus the tangent plane is given by (x, y, f(x, y )) + s(1, 0, x (x, y )) + t(0, 1, y (x, y ) (x + s, y + t, f(x, y ) + x (x, y )s + y (x, y )t) If we replace s by x and t by y, we get our linear approximation of f about (x, y ) ie f(x + x, y + y) f(x, y ) + x (x, y ) x + y (x, y ) y Therefore the above expression states that the tangent plane is a good approximation to the graph When we are working on the tangent plane to the graph of f at (x, y ), we use dx, dy and df These variations on the tangent plane are called differentials Rearranging the last expression, we get f x (x, y ) x + y (x, y ) y, (1) which states that the change f on the graph of f is approximately the change df on the tangent plane This equation in terms of df, dx and dy: df x (x, y )dx + y (x, y )dy, is called the total differential of f at (x, y ) 9

10 We saw that the tangent plane P can be thought of as the graph of the affine mapping (s, t) f(x, y ) + x (x, y )s + y (x, y )t Thus (1) says the change f can be approximated by the linear mapping (s, t) x (x, y )s + y (x, y )t, which we can write in matrix form as ( x (x, y ) y (x, y ) ) ( s t ) Thus we consider the matrix ( x (x, y ) y (x, y ) ) as representing the linear approximation of f around (x, y ) We call this matrix, or the linear map it represents, the (Jacobian) derivative of f at (x, y ) and denote it as Df(x, y ) or Df (x,y ) We can generalize this to functions from R n to R Definition Let f : R n R be a function and let a (a 1,, a n ) R n The total differential of f at point a is df n j1 f j dx j dx dx n The (Jacobian) derivative of f at a, denoted by Df or Df a, is given by ( ) Df Now it is the tangent hyperplane to the n-dimensional graph of f in R n+1 which is a good approximation to the graph itself in the sense that the actual change f is well approximated by the total differential given above with dx i x i Sometimes we write the derivative of f at a as a column matrix: We denote this vector by f or gradf, and call it the gradient (vector) of f at a 10

11 The Chain Rule Sometimes we are interested in how a function changes along a curve in its domain For instance, if inputs are changing with time, we may want to know how the corresponding outputs are changing with time Definition A curve in R n is a function x : R R n given by x(t) (x 1 (t),, x n (t)), for all t R, where each x i : R R is a continuous function The functions x i (t) are called coordinate functions and t is the parameter describing the curve The function x(t) describes the coordinates of the curve at the point where the parameter is t If we think of t as time, then x(t) gives the position of a point on its trajectory in R n at time t If t is time, then x i (t) is the instantaneous velocity of the ith coordinate along the curve at t Definition Let x(t) (x 1 (t),, x n ) be a paremeterized curve in R n The vector x(t) (x 1(t),, x n(t)), is called the velocity vector or the tangent vector of the curve at t Example 10 Consider the curve x(t) y 3, y(t) t 2 When t 2 we are at the point (8, 4) The tangent vector there is (3t 2, 2t) t2 (12, 4) Note (x (0), y (0)) (0, 0) the curve has a cusp at the origin and the tangent vector there is not well-defined We saw in our example that a curve can display irregular behaviour with the possibility of nonsmooth points such as cusps To ensure the existence of a well defined tangent vector at all t, we impose a regularity condition on the curves Definition A curve x(t) is regular if each x i (t) is continuous in t and x (t) 0 for all t Often we want to know how a function f from R n behaves along some regular curve (x 1 (t),, x n (t)), a t b 11

12 y (x (2), y (2)) 4 8 x Figure 6: The parameterized curve (x(t), y(t)) (t 3, t 2 ) The value of the funtion at a point along the curve is given by where g f x : R R g(t) f(x 1 (t),, x 2 (t)), a t b, The derivative g (t) gives the rate of change of f along the curve x(t) Before we state the Chain Rule which tells us how to compute g (t), we need another definition Definition Let U be an open subset of R n and let f : R n R be a function We say f is continuously differentiable or C 1 on U if all its partial derivatives (/ x i ) exist and are continuous for all a U A curve x from an open interval into R n is continuously differentiable (or C 1 ) if each coordinate function x i is continuously differentiable Theorem 7 (Chain Rule I) Let x(t) (x 1 (t),, x n (t)) be a C 1 curve on an open interval about a and let f : R n R be a C 1 function on an open ball about x Then g f x is a C 1 function at a and dg df (x) (x)x dt dt x (x)x 1 x n n Example 11 Let f(x, y) 3x 2 y and let x(t) 2t + 1 and y(t) (t 3) 3 We will compute the total derivative of f with respect to t We could do this directly We have f(x(t)) 3(2t + 1) 2 (t 3) 3, so that df(x(t)) dt 12(2t + 1)(t 3) 3 + 9(2t + 1) 2 (t 3) 2 Using the chain rule, compute / x 6xy, / y 3x 2, x (t) 2 and y (t) 3(t 3) 2 Thus df(x(t)) dt (6xy)2 + (3x 2 )3(t 3) 2 12(2t + 1)(t 3) 3 + 9(2t + 1) 2 (t 3) 3 12

13 It is important to distinguish between the total derivative and the partial derivative Consider a funtion f of three variables x, y, and z Usually we assume these variables are independent, but sometimes they may be dependent on each other y and z, say, could be functions of x In such cases the partial derivative of f with respect to x does not give the true rate of change of f with respect to x, as it does not take account of the dependency of y and z on x The total derivative takes these dependencies into account Example 12 1 Suppose f(x, y, z) xyz The rate of change of f with respect to x is normally found by taking the partial derivative of f with respect to x Here (x, y, z) x yz However, if y and z are not truly independent but depend on x as well this does not give the right answer For a simple example, suppose y x and z x Then f(x, y(x), z(x)) xy(x)z(x) x 3 and so the (total) derivative of f with respect to x is df(x, y(x), z(x)) 3x 2 dx Notice that this is not equal to the partial derivative yz x 2 2 Consider the volume of a cone, which depends on the cone s height h and radius r according to the formula V (r, h) πr2 h 3 The partial derivative of V with respect to r is V r 2πrh 3 It describes the rate with which the cone s volume changes if its radius is varied and its height is kept constant 13

14 The partial derivate with respect to h is V h πr2 3, and represents the rate at which the cone s volume changes if its height is changed and its radius kept constant 2 Now suppose that r(h), or that h(r) Then the total derivatives with respect to r or h are dv dr V r + V dh h dr 2πrh 3 + πr2 3 dh dr dv dh V h + V dr r dh πr πrh dr 3 dh The difference between the total and partial derivatives is the ignorance of indirect dependencies in the latter If, for some reason, the cone s proportions have to stay the same with height and radius in a fixed ratio k, we have k h r dh dr Thus, the total derivative with respect to r is dv dr 2πrh + k πr2 3 3 kπr2 Sometimes we write the chain rule as Compare this with the total differential df dt dx 1 dt + + dx 1 dt We can generalize the chain rule to the case where the inside function depends on several variables Theorem 8 (Chain Rule II) Let x : R s R n, given by x(t) (x 1 (t 1,, t s ),, x n (t 1,, t s )), and f : R n R be C 1 functions Let g f x be the composite function from R s to R Then g is continuously differentiable and g (x) (x) + + (x) t i t i t i t i for all a R s A diagrammatic way to remember the chain rule is given below for the example of a function g : R 2 R given by g(s, t) f(p(s, t), q(t), r(s, t)) 14

15 To find / t (/ s) find the branches ending in t (s) t p p t + dq q dt + r r t s p p s + r r s f p q r Figure 7: Chain Rule II s t t s t Example 13 Suppose u x 2 + 2y, where x r sin(t) and y sin 2 (t) Note that u g(r, t) f(x(r, t), y(r, t)), where f(x, y) x 2 + 2y u r u x x r + u ( y g y r r x x r + ) y y r (2x) sin(t) + 2(0) 2r sin 2 (t) u t u x x t + u y ( y g t t x x t + y (2x)r cos(t) + 2(2 sin(t) cos(t)) 2(r 2 + 2) sin(t) cos(t) Explicit Functions from R n to R m ) y t Until now we have only looked at derivatives of functions with one endogenous variable Often in economics we are interested in functions with several endogenous variables For example, a firm producing m products using n inputs has a production funtion for each output: q 1 f 1 (x 1,, x n ) q 2 f 2 (x 1,, x n ) q m f m (x 1,, x n ) 15

16 We can view the above collection of m functions in n variables as a single function f from R n to R m : f(x) (f 1 (x 1,, x n ), f 2 (x 1,, x n ),, f m (x 1,, x n )) Conversely, if we start with a single funtion f : R n R n as above, we see that each component of f is a function from R n to R Thus it is simple to apply our results for functions from R n to R, such as the chain rule, to the more general case of functions from R n to R m We just apply what we have learnt to each component function f i : R n R and then put it all together in a matrix If, for example, we want to approximate a function f : R n R m (with component functions f 1,, f n ) using differentials, we apply our results to each component f i (see p 324 S&B) We again obtain a matrix of partial derivatives which represents a linear map giving the linear approximation of f about a point a Definition Let f : R n R m be a function The (Jacobian) derivative of f at a, denoted by Df or Df a, is given by 1 1 x Df 2 x 2 2 m m x 2 m This is sometimes called the Jacobian (matrix) An alternative notation is (f 1,, f m ) (x 1,, x n ) When m n 1, we simply have the derivative of a function f : R R and denote it as usual by f Example 14 Suppose, there are two commodities with constant elasticity demand functions q 1 (p 1, p 2, m) 2 p3 2m 2 p 1 and q 2 (p 1, p 2, m) 3 p2 1m p 2 2 in the vicinity of current prices and income (p 1, p 2, m) (2, 4, 1) We want to find out the approximate change in demand for the two goods as a result of a simultaneous change in prices and income 16

17 We totally differentiate each component function q i dq 1 q 1 p 1 dp 1 + q 1 p 2 dp 2 + q 1 m dm ( 2p 2 1 p3 2m 2 )dp 1 + (6p 1 1 p2 2m 2 )dp 2 + (4p 1 1 p3 2m)dm 32dp dp dm at (2,4,1), dq 2 q 2 p 1 dp 1 + q 2 p 2 dp 2 + q 2 m dm (6p 1 p 2 2 m)dp 1 + ( 6p 2 1p 3 2 m)dp 2 + (4p 2 1p 2 2 )dm (3/4)dp 1 (3/8)dp 2 + dm at (2,4,1), Suppose the price of good 1 rises by 01 and the price of good 2 falls by 01 (dp 1 01, dp 2 01) and that income rises by 01 (dm 01) Then dq and dq In matrix notation ( dq1 dq 2 ) ( q1 q 1 q 1 p 1 q 2 p 2 q 2 m q 2 p 1 p 2 m ) dp 1 dp 2 dm So the changes in q 1 and q 2 in the tangent hyperplane at the point (2, 4, 1) are ( ) ( ) dq dq ( ) We can compare this linear approximation to the actual change in the function q (q 1, q 2 ) which can be calculated by substitution The actual change is to three decimal places q ( q 1, q 2 ) ( 7506, 0120) Theorem 9 (Chain Rule III) Let f : R n R m and g : R R n be continuously differentiable functions Let h f g be the composite function from R to R m Then h is continuously differentiable, and for all a R That is h 1 h 2 h m h D(f g) Df(g)g 1 (g) 2 (g) m (g) 1 1 x 2 (g) (g) 2 x 2 (g) 2 (g) m x 2 (g) m (g) g 1 g 2 g n 17

18 The ith component of the above derivative is h i Df i (g) g n j1 i x j (g 1,, g n )g j i (g)g i (g)g n Example 15 Consider the demand functions from the previous example, and suppose now that p 1, p 2 and m vary over time according to the equations p 1 (t) t 2 + 1, p 2 (t) 4t, and m(t) t We want to know the rate of change of demand with respect to time at t 1 First note that (p 1 (1), p 2 (1), m(1)) (2, 4, 1) Therefore ( dq1 dt (1) dq 2 dt (1) ) ( q1 p 1 (p(1)) q 2 p 1 (p(1)) q 1 p 2 (p(1)) q 2 p 2 (p(1)) ( ) ( ) q 1 m (p(1)) q 2 m (p(1)) gives the rate of change of demand over time at t 1 ) p 1(1) p 2(1) m (1) Theorem 10 (Chain Rule IV) Let f : R n R m and g : R s R n be continuously differentiable functions Let h f g be the composite function from R s to R m Then h is continuosly differentiable, and for all a R s Dh D(f g) Df(g)Dg Here Df(g) is an m n Jacobian matrix and Dg is an n s Jacobian matrix The product of these matrices is an m s Jacobian matrix Note that this chain rule is the most general and nests all the other three Writing out the matrices explicitly, the chain rule is: 18

19 h 1 h 1 x s h 2 h 2 x s h m h m x s 1 (g) 1 (g) 2 (g) 2 (g) m (g) m (g) g 1 g 1 x s g 2 g 2 x s g n g n x s Higher Order Derivatives The partial derivative / x i of a function given by f(x 1,, x n )is itself a function of n variables We can continue taking partial derivatives of these partial derivatives Sometimes it is not possible to partially differentiate a function with respect to some variable So we need some terminology describing how smooth functions are Definition Let U be an open subset of R n and let f : R n R be a function We say f is k-times differentiable at a U if all its partial derivatives of order less than k exist If this is true for all a U, we say f is k-times differentiable on U We say f is k-times continuously differentiable or C k at a if all its partial derivatives exist and are continuous at a If this is true for all a U, we say f is k-times continuously differentiable or C k on U There are several types of notation you might see Consider the function y f(x 1,, x n ) For the first order partial derivative, we had the notation x i f i f xi D i f For second order own partial derivatives we have 2 f x 2 i f ii f xix i D ii f For second order cross partial or mixed derivatives we have 2 f x i x j f ij f xix j D ij f 19

20 For higher order partial and mixed derivatives we have r+s+t f x r i xs j xt k Example 16 Consider the Cobb-Douglas utility function u : R 2 u(x, y) 5x 1 5 y 4 5 We will find the second-order derivatives of u First find the first order partial derivatives: u x x 4 5 y 4 5 and u y 4x 1 5 y 1 5 R given by Now find the second order own partial derivatives: 2 u x 2 ( ) u ( ) x y 5 4 x x x 5 x y 5, and 2 u y 2 y ( ) u ( ) 4x 1 5 y y y 5 x 1 5 y 6 5, Now find the second order cross partial derivatives: 2 u y x ( ) u ( ) x y 5 4 y x y 5 x 4 5 y 1 5, and 2 u x y x ( ) u ( ) 4x 1 5 y y x 5 x 4 5 y 1 5, Notice that the function above of two variables has four second order partial derivatives In general, a real-valued function of n variables will have n 2 second order partial derivatives We can array these in a matrix Definition The Hessian (matrix) of a function f : R n R at a point a, denoted by D 2 f or D 2 f a, is given by 2 f 2 f x 2 1 x 2 2 f 2 f D 2 x f 2 2 f 2 f x 2 2 x 2 2 f 2 f x 2 2 f x 2 n It is the n n matrix of cross-partial derivatives Note that the Hessian matrix is the derivative matrix of the vector-valued gradient function f(x), ie D 2 f D[ f(x)] 20

21 In our utility function example we had 2 u y x 2 x y, so that the order of differentiation did not matter It turns out that for functions with continuous second order derivatives, this is always the case Theorem 11 (Young s Theorem) Let U be an open subset of R n and let f : U R be a C 2 function Then D 2 f is a symmetric matrix, ie we have for all i, j 1,, n and for all a U 2 f 2 f x i x j x j x i This means the Hessian is a symmetric matrix, a result you will use when studying demand functions in economics It means that for C 2 utility functions the substitution matrix is symmetric implying that the effect on compensated demand for good j of a rise in the price of good i is the same as the effect on compensated demand for good i of a rise in the price of good j Young s theorem generalizes to the case of taking kth order partial derivatives of C k functions For example, if we take the x 1 x 2 x 4 derivative of order three, then 3 f 3 f 3 f x 2 x 4 x 4 x 2 x 2 x 4 3 f x 2 x 4 3 f x 4 x 2 3 f x 4 x 2 21